{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:KQ4TDE6OWWACQDOAHH6IIKPBMM","short_pith_number":"pith:KQ4TDE6O","canonical_record":{"source":{"id":"1312.2245","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-08T19:20:04Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"38769035e2ea55d3eb6ce51a37d966b83587585e08eb314869020168c263acbd","abstract_canon_sha256":"ed203d795e5e30d8670847695f4de4e8313f0300fd788fbda160b3a0269f3056"},"schema_version":"1.0"},"canonical_sha256":"54393193ceb580280dc039fc8429e16303ddb21316add1d8bffcbe844fb22ad0","source":{"kind":"arxiv","id":"1312.2245","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.2245","created_at":"2026-05-18T03:05:10Z"},{"alias_kind":"arxiv_version","alias_value":"1312.2245v1","created_at":"2026-05-18T03:05:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.2245","created_at":"2026-05-18T03:05:10Z"},{"alias_kind":"pith_short_12","alias_value":"KQ4TDE6OWWAC","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"KQ4TDE6OWWACQDOA","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"KQ4TDE6O","created_at":"2026-05-18T12:27:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:KQ4TDE6OWWACQDOAHH6IIKPBMM","target":"record","payload":{"canonical_record":{"source":{"id":"1312.2245","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-08T19:20:04Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"38769035e2ea55d3eb6ce51a37d966b83587585e08eb314869020168c263acbd","abstract_canon_sha256":"ed203d795e5e30d8670847695f4de4e8313f0300fd788fbda160b3a0269f3056"},"schema_version":"1.0"},"canonical_sha256":"54393193ceb580280dc039fc8429e16303ddb21316add1d8bffcbe844fb22ad0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:10.854097Z","signature_b64":"uCx4Rgg9ej6RVJ2d37eeoF7uVDTN1XOsuvpChBaBT/ZO3xDF/J++vxZAKXb9fouYdNDbmpGmNLXwGsJg9KjdCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"54393193ceb580280dc039fc8429e16303ddb21316add1d8bffcbe844fb22ad0","last_reissued_at":"2026-05-18T03:05:10.853510Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:10.853510Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1312.2245","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:05:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vw6GXAE+bbp8Gys/1y+TCeY4nbugryXYdN5w51WsnwX2rS53sZP4qF82IDmU0fRVjJRHv1F/eWeaR5rpAcbdCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T22:33:50.995165Z"},"content_sha256":"a1a6e278d679e936a3793d4997fe15c77fb9bb975bcebc24ac537a1a562561ed","schema_version":"1.0","event_id":"sha256:a1a6e278d679e936a3793d4997fe15c77fb9bb975bcebc24ac537a1a562561ed"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:KQ4TDE6OWWACQDOAHH6IIKPBMM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Edge-disjoint spanning trees and eigenvalues of regular graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Sebastian M. Cioab\\u{a}, Wiseley Wong","submitted_at":"2013-12-08T19:20:04Z","abstract_excerpt":"Partially answering a question of Paul Seymour, we obtain a sufficient eigenvalue condition for the existence of $k$ edge-disjoint spanning trees in a regular graph, when $k\\in \\{2,3\\}$. More precisely, we show that if the second largest eigenvalue of a $d$-regular graph $G$ is less than $d-\\frac{2k-1}{d+1}$, then $G$ contains at least $k$ edge-disjoint spanning trees, when $k\\in \\{2,3\\}$. We construct examples of graphs that show our bounds are essentially best possible. We conjecture that the above statement is true for any $k<d/2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.2245","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:05:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9cJ3k6mF3zwmmZ3v0PeJrJiN5ZjObzvj8rER6FxDPdUEYvcMCIHTsI21ZsJxZCl5jBwD+1EBxTKoVBi2RrryBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T22:33:50.995608Z"},"content_sha256":"0bc744538a941b7797928e852c0c17637bcf3082fd95441ba795f0b47489017f","schema_version":"1.0","event_id":"sha256:0bc744538a941b7797928e852c0c17637bcf3082fd95441ba795f0b47489017f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KQ4TDE6OWWACQDOAHH6IIKPBMM/bundle.json","state_url":"https://pith.science/pith/KQ4TDE6OWWACQDOAHH6IIKPBMM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KQ4TDE6OWWACQDOAHH6IIKPBMM/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-01T22:33:50Z","links":{"resolver":"https://pith.science/pith/KQ4TDE6OWWACQDOAHH6IIKPBMM","bundle":"https://pith.science/pith/KQ4TDE6OWWACQDOAHH6IIKPBMM/bundle.json","state":"https://pith.science/pith/KQ4TDE6OWWACQDOAHH6IIKPBMM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KQ4TDE6OWWACQDOAHH6IIKPBMM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:KQ4TDE6OWWACQDOAHH6IIKPBMM","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ed203d795e5e30d8670847695f4de4e8313f0300fd788fbda160b3a0269f3056","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-08T19:20:04Z","title_canon_sha256":"38769035e2ea55d3eb6ce51a37d966b83587585e08eb314869020168c263acbd"},"schema_version":"1.0","source":{"id":"1312.2245","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.2245","created_at":"2026-05-18T03:05:10Z"},{"alias_kind":"arxiv_version","alias_value":"1312.2245v1","created_at":"2026-05-18T03:05:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.2245","created_at":"2026-05-18T03:05:10Z"},{"alias_kind":"pith_short_12","alias_value":"KQ4TDE6OWWAC","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"KQ4TDE6OWWACQDOA","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"KQ4TDE6O","created_at":"2026-05-18T12:27:51Z"}],"graph_snapshots":[{"event_id":"sha256:0bc744538a941b7797928e852c0c17637bcf3082fd95441ba795f0b47489017f","target":"graph","created_at":"2026-05-18T03:05:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Partially answering a question of Paul Seymour, we obtain a sufficient eigenvalue condition for the existence of $k$ edge-disjoint spanning trees in a regular graph, when $k\\in \\{2,3\\}$. More precisely, we show that if the second largest eigenvalue of a $d$-regular graph $G$ is less than $d-\\frac{2k-1}{d+1}$, then $G$ contains at least $k$ edge-disjoint spanning trees, when $k\\in \\{2,3\\}$. We construct examples of graphs that show our bounds are essentially best possible. We conjecture that the above statement is true for any $k<d/2$.","authors_text":"Sebastian M. Cioab\\u{a}, Wiseley Wong","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-08T19:20:04Z","title":"Edge-disjoint spanning trees and eigenvalues of regular graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.2245","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a1a6e278d679e936a3793d4997fe15c77fb9bb975bcebc24ac537a1a562561ed","target":"record","created_at":"2026-05-18T03:05:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ed203d795e5e30d8670847695f4de4e8313f0300fd788fbda160b3a0269f3056","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-08T19:20:04Z","title_canon_sha256":"38769035e2ea55d3eb6ce51a37d966b83587585e08eb314869020168c263acbd"},"schema_version":"1.0","source":{"id":"1312.2245","kind":"arxiv","version":1}},"canonical_sha256":"54393193ceb580280dc039fc8429e16303ddb21316add1d8bffcbe844fb22ad0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"54393193ceb580280dc039fc8429e16303ddb21316add1d8bffcbe844fb22ad0","first_computed_at":"2026-05-18T03:05:10.853510Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:05:10.853510Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uCx4Rgg9ej6RVJ2d37eeoF7uVDTN1XOsuvpChBaBT/ZO3xDF/J++vxZAKXb9fouYdNDbmpGmNLXwGsJg9KjdCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:05:10.854097Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.2245","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a1a6e278d679e936a3793d4997fe15c77fb9bb975bcebc24ac537a1a562561ed","sha256:0bc744538a941b7797928e852c0c17637bcf3082fd95441ba795f0b47489017f"],"state_sha256":"158a0d09f1cecbb5f6c22ab51c9870cbf27b7c52fdf3700247c49d296e7577c5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bb/z0ak6Gr0kt8P1E3SRU6Sw19YvBkBk3z2QzVZQlPHtncMxMZ2K8+HsnpEEQyw/TTnra1lcSTA/Q1AC4WbhAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T22:33:50.997559Z","bundle_sha256":"a583e50a5a1ff4f2409b59d87dd46300b11b1742e582a02706267878d290508e"}}